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In mathematics, especially in an area of abstract algebra known as representation theory, a faithful representation ρ of a group ''G'' on a vector space ''V'' is a linear representation in which different elements ''g'' of ''G'' are represented by distinct linear mappings ρ(''g''). In more abstract language, this means that the group homomorphism :ρ: ''G'' → ''GL''(''V'') is injective. ''Caveat:'' While representations of ''G'' over a field ''K'' are ''de facto'' the same as -modules (with denoting the group algebra of the group ''G''), a faithful representation of ''G'' is not necessarily a faithful module for the group algebra. In fact each faithful -module is a faithful representation of ''G'', but the converse does not hold. Consider for example the natural representation of the symmetric group ''S''''n'' in ''n'' dimensions by permutation matrices, which is certainly faithful. Here the order of the group is ''n''! while the ''n''×''n'' matrices form a vector space of dimension ''n''2. As soon as ''n'' is at least 4, dimension counting means that some linear dependence must occur between permutation matrices (since 24 > 16); this relation means that the module for the group algebra is not faithful. ==Properties== A representation ''V'' of a finite group ''G'' over an algebraically closed field ''K'' of characteristic zero is faithful (as a representation) if and only if every irreducible representation of ''G'' occurs as a subrepresentation of ''S''''n''''V'' (the ''n''-th symmetric power of the representation ''V'') for a sufficiently high ''n''. Also, ''V'' is faithful (as a representation) if and only if every irreducible representation of ''G'' occurs as a subrepresentation of : (the ''n''-th tensor power of the representation ''V'') for a sufficiently high ''n''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Faithful representation」の詳細全文を読む スポンサード リンク
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